A ﬂow-based approach to rough diﬀerential equations I. Bailleul

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A flow-based approach to rough differential equations

I. Bailleul

¹

1IRMAR, 263 Avenue du General Leclerc, 35042 RENNES, France,ismael.bailleul@univ-rennes1.fr

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Introduction

This course is dedicated to the study of some class of dynamics in a Banach space, index by time R₊. Although there exists many recipes to cook up such dynamics, those generated by differential equations or vector fields on some configuration space are the most important from a historical point of view. Classical mechanics reached for example its top with the description by Hamilton of the evolution of any classical system as the solution of a first order differential equation with a universal form.

The outcome, in the second half of the twentieth centary, of the study of random phenomena did not really change that state of affair, with the introduction by Itô of stochastic integration and stochastic differential equations.

Classically, one understands a differential equation as the description of a point motion, the set of all these motions being gathered into a single object called aflow.

It is a familly ϕ = ϕts

06s6t6T of maps from the state space to itself, such that ϕtt = Id, for all 0 6 t 6 T, and ϕts = ϕtu◦ϕus, for all 0 6 s 6 u 6 t 6 T. The first aim of the approach to some class of dynamics that is proposed is this course is the construction of flows, as opposed to the construction of trajectories started from some given point.

I will explain in the first part of the course a simple method for constructing a flow ϕ from a family µ= µts

06s6t6T of maps that almost forms a flow. The two essential points of this construction are that

i) ϕts is loosely speaking the composition of infinitely many µti+1ti along an infinite partition s < t1 < · · · < t of the interval [s, t], with infinitesimal mesh,

ii) ϕ depends continuously on µ in some sense.

Our main application of this general machinery will be to study some general class of controlled ordinary differential equations, that is differential equations of the form

dxt=

ℓ

X

i=1

Vi(xt)dhⁱ_t,

where theV_i are vector fields onR^d, say, and the controlshⁱ are real-valued. Giving some meaning and solving such an equation in some general framework is highly non-trivial outside the framework of absolutely continuous controls, without any extra input like probability, under the form of stochastic calculus for instance. It requires Young integration theory for controls with finite p-variation, for16p <2, and Terry Lyons’ theory of rough paths for "rougher" controls! Probabilists are well-acquainted with this kind of situation as stochastic differential equations driven by some Brownian motion are nothing but an example of the kind of problem we

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6 1. INTRODUCTION

intend to tackle. (With no probability!) It is the aim of this course to give you all the necessary tools to understand what is going on here, in the most elementary way as possible, while aiming at some generality.

The general machinery of approximate flows is best illustrated by looking at the classical Cauchy-Lipschitz theory.Fix some Lipschitz continuous vector fieldsVi and some real-valued controls hⁱ of class C¹. It will appear in our setting that a good way of understanding what it means to be a solution to the ordinary differential equation on Rⁿ

(0.1) x˙t=

ℓ

X

i=1

Vi(xt) ˙hⁱ_t=:Vi(xt) ˙hⁱ_t,

is to say that the path x_• satisfies at any time sthe Taylor-type expansion formula xt=xs+ hⁱ_t−hⁱ_s

Vi(xs) +o(t−s), and even

f xt

=f xs

+ hⁱ_t−hⁱ_s Vif

(xs) +O |t−s|² ,

for any function f of class Cb², with Vif standing for the derivative of f in the direction ofVi. Settingµts(x) :=x+ hⁱ_t−hⁱ_s

Vi(x), the preceeding identity rewrites f xt

=f µts(xs)

+O |t−s|² ,

so the elementary map µts provides a very accurate description of the dynamics. It almost forms a flow under mild regularity assumptions on the driving vector fields Vi, and its flow associated by the above "almost-flow to flow" machinery happens to be flow classically generated by equation (0.1).

Going back to a probabilistic setting, what insight does this machinery provide on Stratonovich stochastic differential equations

(0.2) ◦dxt =Vi(xt)◦dwt

driven by some Brownian motion w? The use of this notion of differential enables to write the following kind of Taylor-type expansion of order 2 for any function f of class C³.

f xt

=f xs

+ Z t

s

Vif

(xr)◦dwr

=f xs

+ w_tⁱ−w_sⁱ Vif

(xs) + Z t

s

Z r s

Vj(Vif)

(xu)◦dwu◦dwr

=f xs

(xs) + Z t

s

Z r

s ◦dwu◦dwr

Vj(Vif) (xs) +

Z t s

Z r s

Z u s

(· · ·) (0.3)

For any choice of 2< p <3, the Brownian increments wⁱ_ts :=wⁱ_t−w_sⁱ have almost- surely a size of order(t−s)¹^p, the iterated integralsRt

s

Rr

s ◦dwu◦dwrhave size(t−s)²^p, and the triple integral size (t−s)³^p, with ³_p > 1. What will come later out of this

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1. INTRODUCTION 7

formula is that a solution to equation (0.2) is precisely a path x_• for which one can write for any function f of class C³ a Taylor-type expansion of order 2 of the form f xt

=f xs

(xs)+

Z t s

Z r

s ◦dwu◦dwr

Vj(Vif)

(xs)+O |t−s|^a at any time s, for some exponent a > 1 independent of s. This conclusion puts forward the fact that what the dynamics really see of the Brownian controlw is not only its increments w_ts but also its iterated integrals Rt

s

Rr

s ◦dw_u◦dw_r. The notion of p-rough path X = Xts,X_ts

06s6t6T is an abstraction of this family of pairs of quantities, for 2 < p < 3 here. This multi-level object satisfies some constraints of analytic type (size of its increments) and algebraic type, coming from the higher level parts of the object. As they play the role of some iterated integrals, they need to satisfy some identities consequences of the Chasles relation for elementary integrals:

Rt s =Ru

s +Rt

u. These constraints are all what these rough paths X= (X,X)need to satisfy to give a sense to the equation

(0.4) dx_t =F^⊗(x_t)X(dt)

for a collection F = V1, . . . , Vℓ

of vector fields on Rⁿ, by defining a solution as a path x_• for which one can write some uniform Taylor-type expansion of order 2 (0.5) f xt

=f xs

+X_tsⁱ Vif

(xs) +X^jk_ts Vj(Vkf)

(xs) +O |t−s|^a ,

for any function f of class Cb³. The notation F^⊗ is used here to insist on the fact that it is not only the collection F of vector fields that is used in this definition, but also the differential operators VjVk constructed from F. The introduction and the study of p-rough paths and their collection is done in the second part of the course.

Guided by the results on flows of the first part, we shall reinterpret equation (0.4) to construct directly a flow ϕ solution to the equation

(0.6) dϕ=F^⊗X(dt),

in a sense to be made precise in the third part of the course. The recipe of construction of ϕ will consist in associating to F and X a C¹-approximate flow µ= µts

06s6t6T having everywhere a behaviour similar to that described by equation (0.5), and then to apply the theory described in the first part of the course.

The maps µts will be constructed as the time 1 maps associated with some ordinary differential equation constructed from F and X_ts in a simple way. As they will depend continuously on X, the continuous dependence of ϕ on X will come as a consequence of point ii) above.

All that will be done in a deterministic setting. We shall see in the fourth part of the course how this approach to dynamics is useful in giving a fresh viewpoint on stochastic differential equations and their associated dynamics. The key point will be the fundamental fact that Brownian motion has a natural lift to a Brownianp-rough path, for any2< p <3. Once this object will be constructed by probabilistic means, the deterministic machinery for solving rough differential equations, described in the third part of the course, will enable us to associate to any realization of the Brownian rough path a solution to the rough differential equation (0.4). This solution coincides

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8 1. INTRODUCTION

almost-surely with the solution to the Stratonovich differential equation (0.2)! One shows in that way that this solution is a continuous function of the Brownian rough path, in striking contrast with the fact that it is only a measurable function of the Brownian path itself, with no hope for a more regular dependence in a generic setting. This fact will provide a natural and easy road to the deep results of Wong- Zakai, Stroock & Varadhan or Freidlin & Wentzell.

Several other approaches to rough differential equations are available, each with their own pros and cons. We refer the reader to the books [1] and [2] for an account of Lyons’ original approach; she/he is refered to the book [3] for a thourough account of the Friz-Victoir approach, and to the lecture note [4] by Baudoin for an easier account of their main ideas and results, and to the forthcoming excellent lecture notes [5] by Friz and Hairer on Gubinelli’s point of view. The present approach building on [6] does not overlap with the above ones.¹

1Comments on these lecture notes are most welcome. Please email them at the address ismael.bailleul@univ-rennes1.fr

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CHAPTER 2

Flows and approximate flows

Guide for this chapter

This first part of the course will present the backbone of our approach to rough dynamics under the form of a simple recipe for constructing flows of maps on some Banach space. Although naive, it happens to be robust enough to provide a unified treatment of ordinary, rough and stochastic differential equations. We fix throughout a Banach space V.

The main technical difficulty is to deal with the non-commutative character of the space of maps from V to itself, endowed with the composition operation. To understand the part of the problem that does not come from non-commutativity, let us consider the following model problem. Suppose we are given a family µ =

µts

06s6t61 of elements of some Banach space depending continuously on s and t, and such that

µts

= ot−s(1). Is it possible to construct from µ a family ϕ = ϕts

06s6t61 of elements of that Banach space, depending continuously ons and t, and such that we have

(0.7) ϕtu+ϕus =ϕts

for all 0 6 s 6 u 6 t 6 1? This additivity property plays the role of the flow property. Would the time interval[0,1]be a finite discrete sett1 <· · ·< tn, the additivity property (0.7) would mean thatϕts is the sum of theϕti+1ti, whose definition should beµti+1ti, as these are the only quantities we are given if no arbitrary choice is to be done. Of course, this will not turn ϕ into an additive map, in the sense that property (0.7) holds true, in this discrete setting, but it suggest the following attempt in the continuous setting of the time interval [0,1].

Given a partition π=

0< t1 <· · ·<1 of [0,1] and 06s 6t61, set ϕ^π_ts = X

s6ti<ti+16t

µti+1ti.

This map almost satisfies relation (0.7) as we have

ϕ^π_tu+ϕ^π_us =ϕ^π_ts−µu⁺u⁻ =ϕ^π_ts+o_|π|(1),

for all06s6 u6t 61, whereu⁻, u⁺are the elements ofπsuch thatu⁻ 6u < u⁺, and|π|= max{ti+1−ti}stands for the mesh of the partition. So we expect to find a solutionϕ to our problem under the formϕ^π, for a partition of[0,1]of infinitesimal mesh, that is as a limit of ϕ^π’s, say along a sequence of refined partitions πn where πn+1 has only one more point than πn, say un. However, the sequence ϕ^πⁿ has no reason to converge without assuming further conditions on µ. To fix further the

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10 2. FLOWS AND APPROXIMATE FLOWS

setting, let us consider partitions πn of[0,1]by dyadic times, where we exhaust first all the dyadic times multiples of 2⁻^k, in any order, before taking in the partition points multiples of2⁻^(k+1). Two dyadic timess and t being given, both multiples of 2⁻^k⁰, take n big enough for them to be points of πn. Then, denoting by u⁻_n, u⁺_n the two points of π_n such that u⁻_n < u_n < u⁺_n, the quantity ϕ^π_tsⁿ⁺¹−ϕ^π_tsⁿ will either be null if un∈/ [s, t], or

(0.8) ϕ^π_tsⁿ⁺¹ −ϕ^π_tsⁿ = µ_u⁺_n_u_n+µ_u_n_u⁻_n

−µ_u⁺_n_u⁻_n,

otherwise. A way to control this quantity is to assume that the map µ is approximately additive, in the sense that we have some positive constantsc0 and a >1such that the inequality

(0.9)

µtu+µus

−µts

6c0|t−s|^a holds for all 06s6u6t 61. Under this condition, we have

ϕ^π_tsⁿ⁺¹ −ϕ^π_tsⁿ

6c₀2⁻^am, where

πn+1

= 2⁻^m. There will be2^m such terms in the formal seriesP

n>0 ϕ^π_tsⁿ⁺¹− ϕ^π_tsⁿ

, giving a total contribution for these terms of size 2⁻^(a⁻^1)m, summable in m.

So this sum converges to some quantity ϕts which satisfies (0.7) by construction (on dyadic times only, as defined as above). Note that commutativity of the addition operation was used implicitly to write down equation (0.8).

Somewhat surprisingly, the above approach also works in the non-commutative setting of maps from V to itself under a condition which essentially amounts to replacing the addition operation and the norm | · |in condition (0.9) by the composition operation and the C¹ norm. This will be the essential content of theorem 2 below, taken from the work [6].

1. C¹-approximate flows and their associated flows

We start by defining what will play the role of an approximate flow, in the same way as µ above was understood as an approximately additive map under condition (0.9).

Definition 1. A C¹-approximate flow on V is a familyµ= µ_ts

06s6t6T of C² maps from V to itself, depending continuously on s, t in the topology of uniform convergence, such that

(1.1)

µts−Id

C² =ot−s(1)

and there exists some positive constants c₁ and a >1, such that the inequality

(1.2)

µtu◦µus−µts

C¹ 6c1|t−s|^a holds for all 06s6u6t6T.

Note that µts is required to be C² close to the identity while we ask it to be an approximate flow in aC¹ sense. Given a partitionπts ={s=s0 < s1 <· · ·< sn−1 <

sn =t} of an interval [s, t]⊂[0, T], set

µ_π_ts=µ_t_n_t_n−1 ◦ · · · ◦µ_t₁_t₀.

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1. C¹-APPROXIMATE FLOWS AND THEIR ASSOCIATED FLOWS 11

Theorem 2 (Constructing flows on a Banach space). A C¹-approximate flow defines a unique flow ϕ = ϕts

06s6t6T on V such that the inequality

(1.3)

ϕts−µts

∞6c|t−s|^a

holds for some positive constant c, for all 0 6 s 6 t 6 T sufficiently close, say t−s 6δ. This flow satisfies the inequality

(1.4)

ϕts−µπts

∞ 6 2

1−2¹⁻^ac²₁T πts

a−1

for any partition πts of any interval (s, t) of mesh πts

6δ.

Note that the conclusion of theorem 2 holds in C⁰-norm. This loss of regularity with respect to the controls on µ given by equations (1.1) and (1.2) roughly comes from the use of uniform C¹-estimates on some functions fts to control some increments of the form fts◦gts −fts ◦g_ts^′ , for some C⁰-close maps gts, g_ts^′ . Note that if µ depends continuously on some parameter, thenϕalso depends continuously on that parameter, as a uniform limit of continuous functions, equation (1.8).

The remainder of this section will be dedicated to the proof of theorem 2. We shall proceed in two steps, by proving first that one can construct ϕ as theuniform limit of the µπ’s provided one can control uniformly their Lipschitz norm. This control will be proved in a second step.

1.1. First step. Let us introduce the following inductive definition to prepare the first step.

Definition 3. Let ǫ ∈ (0,1) be given. A partition π = {s = s0 < s1 < · · · <

sn−1 < sn=t} of (s, t) is said to be ǫ-special if it is either trivial or

• one can find an si ∈π sucht that ǫ6 ^s_tⁱ⁻^s

−s 61−ǫ,

• and for any choice u of such ansi, the partitions of [s, u] and[u, t] induced by π are both ǫ-special.

A partition of any interval into sub-intervals of equal length has special type ¹₂. Given a partition π ={s =s0 < s1 <· · ·< sn−1 < sn =t} of(s, t)of special type ǫ and u∈ {s1, . . . , sn−1} with ǫ6 ^u_t⁻^s

−s 61−ǫ, the induced partitions of the intervals [s, u] and [u, t] are also ǫ-special. Set mǫ = sup

ǫ6β61−ǫ

β^a+ (1−β)^a < 1, and pick a constant

L > 2c1

1−m_ǫ,

where c1 is the constant that appears in the dfinition of a C¹-approximate flow, in equation (1.2).

Lemma 4. Let µ = µts

06s6t6T be a C¹-approximate flow on V. Given ǫ > 0, there exists a positive constant δ such that for any 0 6 s 6 t 6 T with t−s 6 δ, and any special partition of type ǫ of an interval (s, t)⊂[0, T], we have

(1.5)

µπts−µts

∞ 6L|t−s|^a.

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Proof – We proceed by induction on the number n of sub-intervals of the partition.

The case n = 2 is the C⁰ version of identity (1.2). Suppose the statement has been proved for n > 2. Fix 0 6 s < t 6 T with t − s 6 δ, and let πts = {s0 = s < s1 < · · · < sn < sn+1 = t} be an ǫ-special partition of [s, t], splitting the interval [s, t]into (n+ 1) sub-intervals. Let u be one of the points of the partition sucht that ǫ6 ^t⁻^u

t−s 61−ǫ, so the two partitions πtu andπus are both −ǫ-special, with respective cardinals no greater thann. Then

µπts−µts

∞ 6

µπtu ◦µπus−µtu◦µπus

∞+

µtu◦µπus−µts

∞

6

µπtu −µtu

∞+

µtu◦µπus−µtu◦µus

∞+

µtu◦µus−µts

∞

6L|t−u|^a+ 1 +oδ(1)

L|u−s|^a+c1|t−s|^a,

by the induction hypothesis and (1.1) and (1.2). Set u−s = β(t−s), with ǫ6β 61−ǫ. The above inequality rewrites

µπts−µts

∞6

n 1 +oδ(1)

(1−β)^a+β^a L+c1

o|t−s|^a.

In order to close the induction, we need to choose δ small enough for the condition

(1.6) c1+ 1 +oδ(1)

mǫL6L

to hold; this can be done since mǫ <1.

As a shorthand, we shall writeµⁿ_ts for ⁿi=0⁻¹µti+1ti, where si =s+_nⁱ(t−s). The next proposition is to be understood as the core of our approach.

Proposition 5 (Step 1). Let µ= µts

06s6t6T be a C¹-approximate flow on V.

Assume the existence of a positive constant δ such that the maps µⁿ_ts, for n >2 and t−s 6δ, are all Lipschitz continuous, with a Lipschitz constant uniformly bounded above by some constant c₂, then there exists a unique flow ϕ = ϕ_ts

06s6t6T on V such that the inequality

(1.7)

ϕts−µts

∞6c|t−s|^a

holds for some positive constant c, for all 06s 6 t 6T with t−s 6 δ. This flow satisfies the inequality

(1.8)

ϕts−µπts

∞6c1c2T πts

a−1

for any partition πts of (s, t), of mesh πts

6δ.

Proof – The existence and uniqueness proofs both rely on the elementary identity (1.9)

fN◦· · ·◦f1−gN◦· · ·◦g1 =

N

X

i=1

gN◦· · ·◦gN−i+1◦fN−i−gN◦· · ·◦gN−i+1◦gN−i

◦fN−i−1◦· · ·◦f1, where the gi and fi are maps from V to itself, and where we use the obvious

convention concerning the summand for the first and last term of the sum. In

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1. C¹-APPROXIMATE FLOWS AND THEIR ASSOCIATED FLOWS 13

particular, if all the maps gN◦ · · · ◦gk are Lipschitz continuous, with a common upper boundc^′ for their Lipschitz constants, then

(1.10)

fN ◦ · · · ◦f1−gN ◦ · · · ◦g1

∞6c^′

N

X

i=1

kfi−gik∞. a) Existence. Set Dδ :=

0 6 s 6 t 6 T; t −s 6 δ and write D_δ for the intersection of Dδ with the set of dyadic real numbers. Given s = a2⁻^k⁰ and t=b2⁻^k⁰ inD_δ, define for n >k0

µ⁽ⁿ⁾_ts :=µ²_tsⁿ =µs_N(n)s_N(n)−1 ◦ · · · ◦µs1s0, where s_i =s+i2⁻ⁿ and s_N(n) =t. Given n >k₀, write

µ⁽ⁿ⁺¹⁾_ts =^N(n)⁻¹

i=0

µ_s_i+1_s_i₊₂⁻ⁿ⁻¹ ◦µ_s_i₊₂⁻ⁿ⁻¹_s_i

and use (1.9) with fi = µ_s_i+1_s_i₊₂⁻ⁿ⁻¹ ◦µ_s_i₊₂⁻ⁿ⁻¹_s_i and gi = µsi+1si and the fact that all the maps µs_N(n)s_N(n)−1◦ · · · ◦µs_N(n)−_i+1s_N(n)−i =µⁱ_s_N(n)_s_N(n)−i are Lipschitz continuous with a common Lipschitz constantc2, by assumption, to get by (1.10) and (1.2)

µ⁽ⁿ⁺¹⁾_ts −µ⁽ⁿ⁾_ts ∞

6c2 N(n)−1

X

i=0

µ_s_i+1_s_i₊₂⁻ⁿ⁻¹◦µ_s_i₊₂⁻ⁿ⁻¹_s_i−µsi+1si

∞6c1c2T 2⁻^(a⁻¹⁾ⁿ;

so µ⁽ⁿ⁾ converges uniformly on D_δ to some continuous function ϕ. We see that ϕ satisfies inequality (1.3) on D_δ as a consequence of (1.5). Asϕ is a uniformly continuous function of(s, t)∈D_δ, by (1.3), it has a unique continuous extension to Dδ, still denoted by ϕ. To see that it defines a flow on Dδ, notice that for dyadic timess6u6t, we have µ⁽ⁿ⁾_ts =µ⁽ⁿ⁾_tu ◦µ⁽ⁿ⁾us, forn big enough; so, since the maps ϕ⁽ⁿ⁾_tu are uniformly Lipschitz continuous, we have ϕts =ϕtu◦ϕus for such triples of times in D_δ, hence for all times since ϕ is continuous. The map ϕ is easily extended as a flow to the whole of {06s6t6T}. Note thatϕ inherits from theµⁿ’s their Lipschitz character, for a Lipschitz constant bounded above by c₂.

b) Uniqueness. Let ψ be any flow satisfying condition (1.3). With formulas (1.9) and (1.10) in mind, rewrite (1.3) under the form ψts =µts+Oc |t−s|^a

, with obvious notations. Then

ψts =ψs2ns2n−1 ◦ · · · ◦ψs1s0 =

µs2ns2n−1 +Oc 2⁻^an

◦ · · · ◦

µs1s0 +Oc 2⁻^an

=µs₂ns₂n−1 ◦ · · · ◦µs1s0+ ∆n=µ⁽ⁿ⁾_ts + ∆n,

where∆nis of the form of the right hand side of (1.9), so is bounded above by a constant multiple of 2⁻^(a⁻¹⁾ⁿ, since all the mapsµs₂ns₂n−1◦ · · · ◦µs₂n−ℓ+1s₂n−ℓ are Lipschitz continuous with a common upper bound for their Lipschitz constants, by assumption. Sending n to infinity shows thatψts =ϕts.

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c) Speed of convergence. Given any partitionπ={s0 =s <· · ·< sn=t}of (s, t), writing ϕts =ⁿi=0⁻¹ϕsi+1si, and using their uniformly Lipschitz character, we see as a consequence of (1.10) that we have for

πts

6δ ϕts −µπts

∞ 6c2 n−1

X

i=0

ϕs_i+1si−µs_i+1si

∞6c1c2 n−1

X

i=0

|si+1−si|^a 6c1c2T πts

a−1

.

Compare what is done in the above proof with what was done in the introduction to this part of the course in a commutative setting.

1.2. Second step. The uniform Lipschitz control assumed in proposition 5 actually holds under the assumption thatµis aC¹-approximate flow. The results of this paragraph could have been proved just after lemma 4 and do not use the result proved in the fundamental proposition 5. Recall L stands for a constant strictly greater than ₁₋^2c_m¹

ǫ.

Proposition 6 (Uniform Lipschitz controls). Let µ = µts

06s6t6T be a C¹- approximate flow on V. Then, given ǫ > 0, there exists a positive constant δ such that the inequality

µ_π_ts−µ_ts

C¹ 6L|t−s|^a

holds for any partition πts of [s, t] of special type ǫ, whenever t−s 6δ.

Proof – We proceed by induction on the number n of sub-intervals of the partition as in the proof of lemma 4. The case n = 2 is identity (1.2). Suppose the statement has been proved for n > 2. Fix 0 6 s < t 6 T with t−s 6 δ, and let πts ={s0 =s < s1 <· · · < sn < sn+1 = t} be an ǫ-special partition of [s, t]

of special, splitting the interval[s, t]into(n+ 1)sub-intervals. Let ube a point of the partition with ǫ 6 ^u_t⁻^s

−s 6 1−ǫ, so that the two partitions πtu and πus

are both ǫ-special, with respective cardinals no greater than n. Then, for any x ∈ V, one can write Dxµπts− Dxµts as a telescopic sum which involve only some controlled quantities.

Dxµπts−Dxµts =Dx µπtu◦µπus

−Dxµts

=

Dµ_πus(x)µπtu−Dµ_πus(x)µtu

Dxµπus

+

Dµ_πus(x)µtu−Dµus(x)µtu

Dxµπus

+ Dµus(x)µtu

Dxµπus−Dxµus

+

Dµus(x)µtu

Dxµus

−Dxµts

=: (1) + (2) + (3) + (4)

We treat each term separately using repeatedly the induction hypothesis, con- tinuity assumption (1.1) for µts inC² topology, and lemma 4 when needed. We first have

(1)

6L|t−u|^a 1 +oδ(1) . Also,

D_µ_πus_(x)µtu−D_µ_us_(x)µtu

6ot−u(1)

µπus(x)−µus(x)

6ot−u(1)L|u−s|^a,

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2. EXERCICES ON FLOWS 15

As the term Dxµπus has size no greater than 1 +oδ(1)

+L|u−s|^a, we have

(2)

6oδ(1)|u−s|^a. Last, we have the upper bound

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6 1 +oδ(1)

L|u−s|^a, while

(4) 6

µtu◦µus−µts

C¹ 6c1|t−s|^a by (1.2). All together, and writing t−u=β(t−s), for some β ∈[ǫ,1−ǫ], this gives

Dxµπts−Dxµts

6

1 +oδ(1)

β^a+ (1−β)^a

L+c1+oδ(1)

|t−s|^a 6L|t−s|^a

for δ small enough, as mǫ <1.

Propositions 5 and 6 together prove theorem 2. Note that an explicit choice of δ is possible as soon as one has a quantitative version of the estimate

µts−Id C² = ot−s(1). Note also that proposition 6 provides an explicit control on the Lipschitz norm of the ϕts, in terms of the Lipschitz norm of µts and L.

2. Exercices on flows

To get a hand on the machinery ofC¹-approximate flows, we shall first see how theorem 2 gives back the classical Cauchy-Lipschitz theory of ordinary differential equations for bounded Lipschitz vector fields on R^d. Working with unbounded Lipschitz vector fields requires a slightly different notion oflocal C¹-approximate flow – see [6].

Theorem 2 can be understood as a non-commutative analogue of Feyel-de la Pradelle’s sewing lemma [7], first introduced by Gubinelli [9] as an abstraction of a fundamental mechanism invented by Lyons [20]. Exercices 2-4 are variations on this commutative version of theorem 2, as already sketched in the introduction to this part.

1. Ordinary differential equations. Let V₁, . . . , V_ℓ be Cb² vector fields on R^d (or a Banach space), and h₁, . . . , h_ℓ be real-valued C¹ controls. Let ϕ stand for the flow associated with the ordinary differential equation

dx_t=V_i(x_t)dhⁱ_t.

a) Show that one defines aC¹-approximate flow setting for all x∈R^d µ_ts(x) =x+ h_t−h_si

V_i(x).

b)Prove that ϕis equal to the flow associated toµby theorem (2). In that sense, a path x is a solution to the above ordinary differential equation if and only if it satisfies at any time sthe Taylor-type expansion formula

f x_t

=f x_s

+ hⁱ_t−hⁱ_s V_if

(x_s) +o(t−s),

for any functionf of classCb². Show that the above reasoning holds true if we only assume that the R^ℓ-valued control h is globally Lipschitz continuous. (It is actually sufficient to supposeh is α-Hölder, for some α > ¹₂.)

c) Does anything go wrong with the above reasoning if the Lipschitz continuous vector fields V_i are not bounded?

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d) Show thatϕ depends continuously onh in the uniform topology for ϕ and the Lipschitz topology forh, defined by the distance

d(h, h^′) =

h₀−h^′₀

+Lip(h−h^′),

where Lip(h−h^′) stands for the Lipschitz norm of h−h^′. (A similar result holds if h is α-Hölder, for someα > ¹₂, with the Lipschitz norm replaced by the α-Hölder norm.)

2. Feyel-de la Pradelle’ commutative sewing lemma. Let V be a Banach space and µ = µ_ts

06s6t61 be a V-valued continuous function. The following commutative version of theorem 2 was first proved under this form by Feyel and de la Pradelle in [7];

see also [8]. Suppose there exists some positive constants c0 and a >1 such that we have

(2.1)

(µ_tu+µ_us)−µ_ts

6c₀|t−s|^a

for all 06s6u6t61. We say the µis an almost-additive functional (or map).

Simplify the proof of theorem 2 to show that there exists a unique mapϕ= ϕ_t

06t61, whose increments ϕ_ts :=ϕ_t−ϕ_s, satisfy

ϕ_ts−µ_ts

6c|t−s|^a for some positive constant cand all06s6t61.

3. Integral products. Letα > ¹₂ be given, and A_t

06t61 be an α-Hölder path with values in the space L_c(V) of continuous linear maps from V to itself. Set A_ts =A_t−A_s, for 06s6t61, and use the notation | · | for the operator norm on Lc(V).

a)Use theorem (2) to show that setting µ_ts=Id+A_ts determines a unique flowϕ on V. A good notation forϕ_ts isQ

s6r6t Id+dA_r

=Q

s6r6te^dA^r. b)Let B_t

06t61 be another L_c(V)-valued α-Hölder path. Show that we define a C¹-approximate flow µ setting µts = Id+Ats

Id+Bts

. Prove as a consequence the well-known formula

e^A⁰^+B⁰ = lim

n

e^A²ⁿ⁰e^B²ⁿ⁰2ⁿ

.

4. Controls and finite-variation paths. A control is a non-negative map ω = ω_ts

06s6t61, null on the diagonal, and such that we have ω_tu+ω_us 6ω_ts for all 06s6u6t61.

a)Show that Feyel and de la Pradelle’ sewing lemma holds true if we replacet−s by ω_ts, if we suppose that ω^a is a control.

b)Recall that a V-valued path x= x_t

06t61 is said to have finite p-variation, for some p>1, if the following quantity is finite for all06s6t61:

|x|^p_p₋var;[s,t]:= supX

x_t_i+1−x_t_i

p,

with a sum over the partition points t_i of a given partition of the interval [s, t], and a supremum over the set of all partitions of [s, t]. Such a definition is invariant by any reparametrization of the time interval [s, t]. Given such a path, show that setting ωts =

|x|^p_p₋var;[s,t], defines a controlω.

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2. EXERCICES ON FLOWS 17

c)Show that a path with finite p-variation can be reparametrized into a ¹_p-Hölder path, with 1-Hölder paths being understood as Lipschitz continuous paths. Given an R^ℓ-valued pathh with finite 1-variation, set

ζ_s= inf{t>0 ; |h|1−var;[0,t]>s}. We define a solutionx_• to the ordinary differential equation

dxt=Vi(xt)dhⁱ_t

driven by has a path x_• such that the reparametrized path y:=x◦ζ is a solution to the ordinary differential equation

dy_s=V_i(y_s)d(h◦ζ)ⁱ_t, driven by the globally Lipschitz path h◦ζ.

d) Prove that the flow ϕconstructed in this case from theorem 2 depends continuously on h in the uniform norm for ϕ and the 1-variation topology associated with the norm|·|1−varforh. (Following the remarks of exercice 1, one can actually prove the results of questions c) and d) for paths with finite p-variation, for16p <2.)

5. Young integral. Given another Banach space E, denote by L_c(V,E) the space of continuous linear maps from V to E equipped with the operator norm. Let α and β be positive real numbers such that α+β > 1. Given any 0< α <1, we denote by Lip_α(E) the set of α-Hölder maps. This unusual notation will be justified in the third path of the course.

a) Given an L_c(V,E)-valued α-Lipschitz map x = x_s

06s61 and a V-valued β- Lipschitz map y= y_s

06s61, show that setting µ_ts =x_s y_t−y_s

for all 06s 6t61, defines an E-valued function µ that satisfies equation (2.1), with a constant c₀ to be made explicit.

b)The associated function ϕ is denoted by ϕt=Rt

0 xsdys, for all 06t61. Show that it is a continuous function ofx∈Lip_α Lc(V,E)

andy∈Lip_β(V).

6. Lipschitz dependence of ϕ on µ. As emphasized in the remark following theorem 2, inequality (1.8) implies that ϕ, understood as a function of µ, is continuous in the C⁰-norm on the sets of µ’s of the form

µ;(1.2) holds uniformly , equipped with the C⁰-norm. One can actually prove that it depends Lipschitz continuously on µ in the following sense.

Let µ = µ_ts

06s6t61 and µ^′ = µ^′_ts

06s6t61 be C¹-approximate flows on V, with associated flows ϕand ϕ^′. Suppose that we have

µ_ts−µ^′_ts

C¹ 6ǫ|t−s|¹^p and

µ_tu◦µ_us−µ_ts

− µ^′_tu◦µ^′_us−µ^′_ts

C¹ 6ǫ|t−s|^a

for a positive constant ǫ, witha >1as in the definition of theC¹-approximate flowsµ, µ^′, for all 06s6u6t61. Prove that one has

ϕ_ts−µ_ts

− ϕ^′_ts−µ^′_ts

∞6c ǫ|t−s|^a, for all for 06s6t61, for some explicit positive constant c.

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CHAPTER 3

Rough paths

Guide for this chapter

Hölder p-rough paths, which control the rough differential equations dxt=F(xt)X(dt), dϕt=F^⊗X(dt),

and play the role of the controlhin the model classical ordinary differential equation dxt =Vi(xt)dhⁱ_t=F(xt)dht

are defined in section 1.2. As R^ℓ-valued paths, they are not regular enough for the formula

µts(x) =x+X_tsⁱ Vi(x)

to define an approximate flow, as in the classical Euler scheme studied in exercice 1.

The missing bit of information needed to stabilize the situation is a substitute of the non-existing iterated integrals Rt

s X_r^jdX_r^k, and higher order iterated integrals, which provide a partial description of what happens to X during any time interval (s, t).

A (Hölder) p-rough path is a multi-level object whose higher order parts provide precisely that information. We saw in the introduction that iterated integrals appear naturally in Taylor-Euler expansions of solutions to ordinary differential equations;

they provide higher order numerical schemes like Milstein’ second order scheme. It is an important fact thatp-rough paths take values in a very special kind of algebraic structure, whose basic features are explained in section 1.1. A Hölder p-rough path will then appear as a kind of ¹_p-Hölder path in that space. We shall then study in section 2 the space of p-rough path for itself.

1. Definition of a Hölder p-rough path Iterated integrals, as they appear for instance under the form Rt

s

Ry

s dh^j_rdh^k_u or Rt

s

Ry s

Rr

s(· · ·), are multi-indexed quantities. A useful formalism to work with such objects is provided by the notion of tensor product. We first start our investigations by recalling some elementary facts about that notion. Eventually, all what will be used for practical computations on rough differential equations will be a product operation very similar to the product operation on polynomials. This abstract setting however greatly clarifies the meaning of these computations.

19

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20 3. ROUGH PATHS

1.1. An algebraic prelude: tensor algebra over R^ℓ and free nilpotent Lie group. Let first recall what the algebraic tensor product U⊗V of any two Banach spaces U and V is. Denote by V’ the set of all continuous linear forms on V. Given u∈U and v ∈V, we define a continuous linear map on V’ setting

(u⊗v)(v^′) = (v^′, v)u,

for any v^′ ∈ V^′. The algebraic tensor product U⊗V is the set of all finite linear combinations of such maps. Its elementary elements u⊗v are 1-dimensional rank maps. Note that an element of U⊗V can have several different decompositions as a sum of elementary elements; this has no consequences as they all define the same map from V’ to U.

As an example, R^ℓ ⊗(R^ℓ)^′ is the set of all linear maps from R^ℓ to itself, that is L(R^ℓ). We keep that interpretation for R^ℓ⊗R^ℓ, as R^ℓ and (R^ℓ)^′ are canonically identified. To see which element of L(R^ℓ)corresponds tou⊗v, it suffices to look at the image of thej^th vectorǫ_j of the canonical basis by the linear map u⊗v; it gives the j^th column of the matrix of u⊗v in the canonical basis. We have

(u⊗v) ǫj

= (v, ǫj)u.

The family ǫi1 ⊗ · · · ⊗ǫik

16i1,...,ik6ℓ defines the canonical basis of (R^ℓ)^⊗^k.

•ForN ∈N∪{∞}, writeT_ℓ^(N)for the direct sum L^N

r=0

R^ℓ_⊗r

, with the convention that R^ℓ_⊗0

stands forR. Denote by a= ⊕^N

r=0a^r and b= ⊕^N

r=0b^r two generic elements of T_ℓ^(N). The vector space T_ℓ^(N) is an algebra for the operations

a+b= ⊕^N

r=0(a^r+b^r), ab= ⊕^N

r=0c^r, with c^r =

r

X

k=0

a^k⊗b^r⁻^k ∈(R^ℓ)^⊗^r (1.1)

It is called the (truncated) tensor algebra of R^ℓ (if N is finite). Note the similarity between these rules and the analogue rules for addition and product of polynomials.

The exponential map exp : T_ℓ⁽^∞⁾ → T_ℓ⁽^∞⁾ and the logarithm map log : T_ℓ⁽^∞⁾ → T_ℓ⁽^∞⁾ are defined by the usual series

(1.2) exp(a) =X

n>0

aⁿ

n!, log(b) =X

n>1

(−1)ⁿ

n (1−b)ⁿ,

with the convention a⁰ = 1 ∈R ⊂ T_ℓ⁽^∞⁾. Denote by πN :T_ℓ⁽^∞⁾ → T_ℓ^(N⁾ the natural projection. We also denote by exp and log the restrictions to T_ℓ^(N) of the maps π_N ◦exp and π_N ◦log respectively. Denote by T_ℓ^(N^),1, resp. T_ℓ^(N),0, the elements a0 ⊕ · · · ⊕cN of T_ℓ^(N) such that a0 = 0, resp. a0 = 1. All the elements of T_ℓ^(N),1 are

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1. DEFINITION OF A HÖLDER p-ROUGH PATH 21

invertible, and exp :T_ℓ^(N),0 →T_ℓ^(N^),1 andlog :T_ℓ^(N^),1 →T_ℓ^(N),0 are smooth reciprocal bijections. As an example, with a= 1⊕a¹⊕a² ∈T_ℓ^2,1, we have

loga= 0⊕(−a¹)⊕1

2a¹ ⊗a¹−a² .

The set T_ℓ^(N^),1 is naturally equipped with a norm defined by the formula

(1.3) kak:=

ℓ

X

i=1

aⁱ

1 i

Eucl, where

aⁱ

_Eucl stands for the Euclidean norm of aⁱ ∈ (R^ℓ)^⊗ⁱ, identified with an element of R^ℓⁱ by looking at its coordinates in the canonical basis of (R^ℓ)^⊗ⁱ. The choice of power ¹_i comes from the fact that T_ℓ^(N),1 is naturally equipped with a dilation operation

(1.4) δλ(a) = 1, λa¹, . . . , λ^Na^N ,

so the norm k · k is homogeneous with respect to this dilation, in the sense that one has

δ_λ(a)

=|λ|kak for all λ ∈R, and all a∈T_ℓ^(N),1.

The formula [a,b] =ab−ba, defines a Lie bracket on T_ℓ^(N). Define inductively f:=f¹ :=R^ℓ, considered as a subset ofT_ℓ⁽^∞⁾, andfⁿ⁺¹ = [f,fⁿ]⊂T_ℓ⁽^∞⁾.

Definition 7. • The Lie algebra g^N_ℓ generated by the f¹, . . . ,f^N in T_ℓ^(N) is called the N-step free nilpotent Lie algebra.

• As a consequence of Baker-Campbell-Hausdorf-Dynkin formula, the subset exp g^N_ℓ

of T_ℓ^(N),1 is a group for the multiplication operation. It is called the N-step nilpotent Lie group on R^ℓ and denoted by G^N_ℓ .

As all finite dimensional Lie groups, the N-step nilpotent Lie group is equipped with a natural (sub-Riemannian) distance inherited from its manifold structure. Its definition rests on the fact that the element au of T_ℓ^(N) is for any a ∈ G^(N)_ℓ and u ∈ R^ℓ ⊂ T_ℓ^(N) a tangent vector to G^(N_ℓ ⁾ at point a (as u is tangent to G^(N_ℓ ⁾ at the identity and tangent vectors are transported by left translation in the group). So the ordinary differential equation

da_t =a_th˙_t

makes sense for any R^ℓ-valued smooth controlh, and defines a path in G^(N)_ℓ started from the identity. We define the size |a| of a by the formula

|a|= inf Z 1

0

h˙t

dt,

where the infimum is over the set of all piecewise smooth controlshsuch thata₁ =a. This set is non-empty as a ∈ exp g^N

can be written as a₁ for some piecewise C¹ control, as a consequence of a theorem of sub-Riemannian geometry due to Chow;

see for instance the textbook [11] for a nice account of that theorem. The distance

A ﬂow-based approach to rough diﬀerential equations I. Bailleul

A flow-based approach to rough differential equations

I. Bailleul

Contents

Introduction

Flows and approximate flows

Rough paths